Question

(q²-p²)(q²+p²)-(q²-p²)²​

Answer 1

QUESTION :-

Factorize : $(q^2 - p^2)(q^2 + p^2) - (q^2 - p^2)^2$

SOLUTION :-

$[(q^2 - p^2)(q^2 + p^2)] - (q^2 - p^2)^2$

Using the Identity, $(a + b)(a - b) = a^2 - b^2$,

$\Rightarrow [(q^2 - p^2)(q^2 + p^2)] - (q^2 - p^2)^2 = (q^2)^2 - (p^2)^2 - (q^2 - p^2)^2\\\\\Rightarrow \bold {q^4 - p^4 - [(q^2 - p^2)^2]}$

Using the Identity, $(a - b)^2 = a^2 - 2ab + b^2$,

$\Rightarrow q^4 - p^4 - [(q^2 - p^2)^2] = q^4 - p^4 - [(q^2)^2 - 2(q^2)(p^2) + (p^2)^2\\\\\Rightarrow q^4 - p^4 - [q^4 - 2q^2p^2 + p^4]\\\\\Rightarrow q^4 - p^4 - q^4 + 2q^2p^2 - p^4\\\\\Rightarrow - p^4 + 2q^2p^2 - p^4} = \bold {-2p^4 + 2q^2p^2}\\\\\Rightarrow \bold {- [2p^4 - 2q^2p^2]} = \bold {-[2p^2 (p^2 - q^2)]}$

Using the Identity, $a^2 - b^2 = (a + b)(a - b)$,

$\bold {-[2p^2 (p^2 - q^2)]} = \bold {-[2p^2 (p + q)(p - q)]}$

CONCLUSION :-

The Factorized Form of $(q^2 - p^2)(q^2 + p^2) - (q^2 - p^2)^2$ is $\bold {-[2p^2 (p + q)(p - q)]}$